J. of Inequal. & Appl., 1997, Vol. 1, pp. 223-238
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On Weighted Hardy Inequalities on
Semiaxis for Functions Vanishing
at the Endpoints
MASHA NASYROVAa and VLADIMIR STEPANOVb,*,t
Khabarovsk State University, Department of Applied Mathematics,
Tichookeanskaya, 136, Khabarovsk, 680035, Russia
b
Computer Center of the Far-Eastern Branch of the Russian Academy
of Sciences, Shelest 118-205, Khabarovsk, 680042, Russia
E-mail: 1600 @ as.khabarovsk.su
(Received 15 July 1996)
We study the weighted Hardy inequalities on the semiaxis of the form
for functions vanishing at the endpoints together with derivatives up to the order k
k 2 is completely characterized.
1. The case
Keywords: Weighted Hardy’s inequality.
1991 Mathematics Subject Classification: Primary 26D 10; Secondary 34B05, 46N20.
1 INTRODUCTION
In the theory of the weighted Hardy inequalities for higher-order derivatives
the problem, when functions vanish together with derivatives at the endpoints,
is still undecided. The case k 1 has been solved by P. Gurka 1 and activity
in this area has increased in recent years mostly due to the efforts of A. Kufner,
*Author for correspondence.
?The research work of the second author was partially supported by INTAS project grant
94-881 and EPSRC grant GR-K76344 of Great Britain.
223
M. NASYROVA and V. STEPANOV
224
who contributed to various other cases by himself and with coauthors as well
[2-6].
We study the problem on the semiaxis, which is different in some aspects
from the case of a finite interval [3,9] since the inequality (1), k > 1, becomes
non-trivial with only one condition F (cx) 0. We give a complete solution
for the ase k 2 and hope that our approach can be extended to the general
situation on the semiaxis.
The inequality (1), k > 1, with the boundary conditions
O, F (j) (oc)
F (i) (0)
for some 0 < i, j < k
0
(2)
1
becomes non-trivial, if the right-hand side of (1) for all functions, having
absolutely continuous derivative F (k- 1) (x) and satisfying (2), is a norm in a
corresponding weighted space. We suppose throughout the paper, that u (x) as
well as v(x) and Iv(x)1-1 be locally square integrable; the last is necessary and
sufficient for the condition, that any function F satisfying F(k)v < cxz,
can be represented by a Riemann-Liouville integral.
Let o/=(o/0, o/1
k- 1; lal-0/k-l), o/j-0, 1; j=0, 1
112
Put
F (’) (0)
0
F () ((x)
.
O<j<k-1
F (j) (0)
F (i) (cxz)
0
for all j, when o/j
0
0
for all i, when
fli
1,
1
and
I’2
A C (O/
2
F() (0)
F () (oc)
0}.
For a finite interval and k > 1 the boundary value problem
F () (x)
0, F (a) (0)
0, F () (a)
(3)
0
has a non-trivial solution, if 1 < lal / I1 _< k 1. Hence, it might have
only the trivial solution, only if lal / I1 >_ k. On the other hand, (3) has
only the trivial solution, if la1-4- I1 >_ 2Ck 1) while if lal / I1 >_ 2k, then
functions in
functions belonging to
(a, ) can be approximated by
the norm of ACe (o/, ). Thus, on a finite interval we have
AC
Nk
C
22k- -t
(2k)!
2(k !)2
WEIGHTED HARDY INEQUALITIES
225
ACke(ot, fl), k
IIFIIAc(,,) is not a
< I1 + I1
2k; however, for some of them
norm. The last number is given by rather more
complicated formulae. The situation on (0, cx) is slightly different and
affected by the following observation. Let (1) holds and fl(1) and fl2) be
multiindices such, that min{i" jl)
1}
1}. Then
min{i"
fl(2)). It implies an heuristic principle, that the
ACk2(ot, fl(1))
characterization problems of (1) for functions with the boundary couples
(or, fl(1)) and (or, fl(2)) are equivalent. Similar effects take place on (-x, 0)
or (-x, x).
For simplicity we restrict ourselves to case of L2-norm, however the
methods of the present paper in conjunction with the results 12-14] provide
the Lp-Lq case as well.
Throughout the paper uncertainties of the form 0.c, 0/0, x/x are taken
equal to zero, the inequality A << B means A < cB with an absolute constant
B is
c, perhaps, different in different places, however the relationship A
interpreted as A << B << A or A cB. XE denotes the characteristic function
of a set E.
spaces
fl2)
AC(ot,
2 REMARKS ON GENERAL CASE
If k > l, then the following characterization are known.
(1)
(2)
(3)
(4)
(5)
k 1. Io1 / I/1 1 [7], I1 / Itl 2 [1].
k > 1. Icl k, I1 0 or I1 0, I1 k [10].
k > 1. I1 / I/1 k, max{j otj 1} + 1 min{i fli 1} [2].
k > 1. Finite interval, Icl / I1 k [9].
1 and the
k > 1. Finite interval, Icl / I1 k / 1, Ok/ /k/l
remaining parts of multiindices satisfy the P61ya condition [4-5].
_
Below (Theorem 1.2) we supplement this list by the following
(6) k > 1. Infinite interval (a, x), 1 < Io1 / I1 < k, I1 >_ 1,
j 0, 1
Io1, I1 / 1 min{i /i 1}.
Let (a, b)
1,
(-cx, o), a < b and k > 1. We need the following notations.
Le, v
Ikf (x)
j
L2,v,(a,b)--
lk,(a,b)f (x)
[if" II/vll
1
1-’(k)
(x
<
y)-I f(y) dy,
x e (a, b),
M. NASYROVA and V. STEPANOV
226
Jkg(X)
e,o
ek,o;(a,b),u v2
2
e k,1
e 2k,l’(a,b),u,v
n,
B,
I"(k)
(y
(X
sup
x)k-l g(y) dy,
x
(a, b),
t)2(k-1)lu(x)12 dx
Iv1-2,
a<t<b
lul 2
sup
x)2(k-1)lv(x)1-2dx
(t
a<t<b
2
Bk,O;(a,b),u,v
0
if b
Jk,(a,b)g(x)
sup
a<t<b
Bk,2 1" (a,b),u,v
fa
b
12f Iv1-2
x )2(k-l) lu(x) dx
(t
lu[ 2
sup
t)2(l-l)lv(x)l-2 dx
(x
a<t<b
a
a;(a,b),u,v max(a2,0, a2k,1),
S B;(a,b),u, v max(B, B,I).
0,
Also we apply throughout the paper the following.
THEOREM 1.1 [10--12] Necessary and sufficient conditions for the inequality
(1) to be valid on interval (a, b) are:
(1) At < cx, when F(a)- F’(a) ...-- F(k)(a)= OandthenC A,
F () (b) 0 and then C , Bk.
(2) Bk < o, when F(b) F f(b)
Our first result is the following simple observation.
THEOREM 1.2 Let -x < a < cxz. Necessary and sufficient conditions for
the inequality (1) to be validon (a, cxz), when F(cxz) O, is B;(a,),u,v < cxz
and then C Bk;(a,),u,v.
,
Necessity follows from Theorem 1.1. For sufficiency suppose
0 and Bk;(a,),u,v < x. Denote F (k)
< x, F(x)
f
Proof
[[F(k)vl[2
and let
P(x)
Since
Jf(x)
(Y
r(k)
x)k-1 f(Y) dy,
x > a.
Bk;(a,c),u,v < cx, we see, that
IF(x)l
1
< --llfvll2
and conclude, that F
sufficiency.
.
(fx
x
(y
x)e(-l)[v(y)[-2 dy
)1/2
O, x
Applying Theorem 1.1 again, we obtain the
WEIGHTED HARDY INEQUALITIES
227
Remark 1.1 Theorem 1.2 implies the limiting case of the assertion (6)
1. The remaining part follows similarly by
above, when I1
0, I1
application of the result of [2]. The situation on the left semiaxis is similar,
however, for the real line the only limiting cases of (6), that is Io1
0,
0
or
0
valid.
are
0,
(1,
(1,
),
fl
I11
3 THE CASEk=2
In this section we consider the inequality
(3)
on (0, oo). The full list of non-trivial cases is the following.
Icl + I/1
I1 + I/1
I1 + I/1
Icl + I/1
1. (1.1) F(oo) --0.
2. (2.1) F(0) F(oo) --0,
(2.2) V(0) V’(0) 0,
(2.3)
(2.4)
(2.5)
3. (3.1)
(3.2)
(3.3)
(3.4)
4. (4.1)
F(oo) F’(oo) 0,
V(0)= V’(oo) 0,
F’(0) F(oo) 0.
F(0) F’(0) V(oo) 0,
F(0) F’(0) F’(oo) 0,
F(0) F(oo) V’(oo) --0,
F’(0) F(oo) F’(oo) 0.
F(0) F’(0) F(oo) F’(oo)
0.
from
The case (1.1) follows from Theorem 1.2, (2.2) and (2.3)
Theorem 1.1, (2.4) from [2]. Applying the heuristic principle we show
that
(1.1)
(2.3), (2.1)
(3.3), (2.5)
(3.4), (3.1)
(4.1).
3 need to be treated.
Thus, the only four cases of subsection lal + I11
Moreover, we show that the cases (3.2), (3.3) and (3.4) are similar to each
other and begin with the first of them.
CASE (3.2). It is easy to see, that the characterization problem
IIFull2 _< C IIv" ll , F(0)= F’(0)= F’(oo)= 0
M. NASYROVA and V. STEPANOV
228
is equivalent to the following
< Cllfvll2,
ll(/2f) ull2
Let r
6
f --O.
(4)
(0, c) be defined by
fo Iv1-2 f Iv1-2
(5)
and put
lul 2
THEOREM 2.1
C
(z-
X) 2 Iv(x)1-2 dx.
The leastpossible constant C in (4) is sandwiched as follows
,, A2;(o,r),u,v + Dr + A1;(r,o),u,(x_r)-lv(x) + B;(r,c),x-r)ux),v.
Proof
If
fo f
(6)
0, then for x > r we have
’,-fo (fosts-fo (o )s+f (fots_
fo
fo
(r
(r
fX(fs f)
y) f (y)dy
y) f (y)dy
(x
r)
1 fx
f
ds
r) f (y)dy.
(y
(7)
For the upper bound we use Theorem 1.1 and Cauchy’s inequality and find
r(r
y)f(y)dy
x
X[r’,](x)(x
r)u(x)
X[r’,](x)u(x)
f
(y
r) f (y)dy
2
(A2;(o,r’),u,v + Dr, h- A1;(r’,o),u,(x-r)-,v(x) q- B;(r’,),(x-r)u(x),v) Ilfvll2.
WEIGHTED HARDY INEQUALITIES
229
For the lower bound it is sufficient to arrange a one-to-one isometrical map
or between the cones
1
{f
L2,v
f >_. O, supp f c_ [0, z]}
and
/22={fL2,v f<0, supp f___[z, cx])
such that
(f q- cot f)
0,
f
(8)
We construct cot below, using an idea from ([8], Section 1.8). Now, suppose
(4) is tree and the right-hand side of (5) is finite. Since for any function of
the form f -+- ogr f, f /21 or o) f + f, f /22 the right-hand side of (7)
is nonnegative, we obtain the following inequalities
[IX[O,r] (I2f)ul[ 2 < ll(I2(f + rf)) ull2 C II(f + rf)vll2
2C Ilxt0,foll, f c
and
IIt,a (I2f u[12
2c
[[t,afllz, f
Then the required lower bound follows from Theorem 1.1. For a possible
infinite right hand side of (5) we proceed as before, replacing the weight
0.
v(x) by (1 + ex)v(x) and then using limiting guments with
To construct the map wr we define for f 6 1
(-i()[ (-, <)i
(wrf)(x)
where p
[0, r]
Iv(x)l 2
Iv, ] is given by
Iv1-2,
Iv
s e [0,
(s)
Then (8) is valid d
t, f
Theorem 2.1 is proved.
[2
t,
< f [[2’
f
x
v,
M. NASYROVA and V. STEPANOV
230
Theorem 2.1 has a complete analog for a finite interval,
(-cx, 0) or (-cx, cxz) and for an Lp-Lq setting. For a finite interval the
Lp-Lq version of Theorem 2.1 follows from the characterization (6) above,
which has independently been proved by a different method in [5].
Remark 2.1
CASE (3.4). In this case the characterization of
C
IIFull2
IIF" l12,
F’(0)- F(cx)- F’()= 0
is equivalent to the problem
II(J2g) ul12 < C Ilgvll2,
g
(9)
0,
which is dual to (4). Put
D 2a,r
u
12
(x
z’) 2 Iv(x)1-2 dx.
THEOREM 2.2 The least possible constant C in (9) is estimated by
B2;(r,oo),u,v + Dl,r -+- A1;(o,r),(r-x)u(x),v -+- B1;(O,r),u,(r-x)-lv(x) (10)
C
Proof
This time we use a decomposition for 0 < x < r of the form
Jg(x)
(y
r)g(y) dy
(
x)
g
(r
y)g(y) dy
and proceed as in the proof of Theorem 2.1.
CASE (3.3). Now we have
Ilfull2 _< C IIf’ l12, f(0)- F(cz)- F’(oc)= 0,
which equivalent to the problem
II(Jg) ull < C Ilgvll
y g(y) dy
O.
Making the substitution y g(y) -+ g(y), we make (11) equivalent to
(11)
WEIGHTED HARDY INEQUALITIES
where vl (x)
231
-v(x) and
J2g(x)
1
g(y) dy.
--f
By a known criterion 14]
2211L2,vl,(r,c)-+
L2,u,(r,c)
L2,v,(r,ee)
L2,u,(r,)
Thus, using the decomposition
x
-2g(x)
dy
(y- "c)g(y)
y
z-x
fr
r.
foX Xfxr
g
dy
y)g(y), 0 < x < r
y
(3
and arguing as in the proof of Theorems 2.1 and 2.2 we obtain
THEOREM 2.3
C
The least constant C in (11) is estimated by
B2;(r,),u,v -+- z"-1
(D2,r -t- A1;(O,r),(r-x)u(x),x-lv(x)
-’l-Bl.(O,r),xu(x),(r-x)-Iv(x))
(12)
where
2
D2,r
f0 x2lu(x)l
2
dx
(x
z-)2lv(x)1-2 dx
CASE (3.1). Now we have equivalence of
IlFull2
C
[[F"v[[ 2
F(0)
F’(0)
F(cx)
0
and
y
II(lzf) ullz _< C Ilfvll2,
f0c(f0 f)
dy
where the integral is interpreted in the Riemann sense.
We need the following simple note.
O,
(13)
M. NASYROVA and V. STEPANOV
232
A locally integrable function f satisfies
LEMMA
fo(foY f)
(14)
0
dy
(0, cxz) such, that
if and only if there exist )
z
y
foZf--O fo (fZf)dy-- fz (fz f)
and
dy.
(15)
Proof Suppose (14) be valid, then the first assertion in (15) is trivial. Then
we have
O--
ji (foYf) dY-- foL (foYf)
-ji’ (fy’f)
dy-k-
dy-t-
fc (foYf)
f (fYf)
dy-
dy.
Conversing the last line, we prove the "if" part.
Suppose ) 6 (0, c) and put
f0
vz
THEOREM 2.4
C
101-2
101-2.
The least possible constant C in (13) is given by
A2;(0,c),u,v + sup (Al’(rz,Z),u,(x-rz)-’v(x)
+ Bl’(rz,.),(x-r)u(x),v).
)>0
Proof We begin with the upper bound. Let f
Hence, (15) is true for some )
I2 f (x)
fo
(;k
6
L2,v and (14) be valid.
(0, cx). Then for x >
y) f (y) dy
+
(x
.
(16)
we have
y) f (y) dy.
Using this and applying Theorems 1.1 and 2.1, we obtain
WEIGHTED HARDY INEQUALITIES
sup
A2;(O,r),u,v +
x)21v(x)1-2 dx+
(’cz
lu
233
A1; (rx,Z),u, (x-rx)-l v(x) -t-
lul 2
(z
x)21v(x)1-2 dx + Aa;(x,,,
- f0
For the lower bound we suppose ;
[rz, )] by
Pl [0, rz]
Given f
L2,v, supp
outside [:, )] and
Iv1-2-
l(s)
6
Iv1-2
f c_ [0, rz], f(x)
f (x)
(0, x) and define the function
s[0
> 0 we define fl(x) to be zero
x
Iv(x)l
(rx, .).
Then, by the argument from the proof of Theorem 2.1 we have
f+
Now, given )
(0, cx) we find IX
f0 x21v(x)1-2
and define the function/92
fo x2lv(x)1-2
dx
fl
IX())
(Ix
0.
(17)
(0, cx) from equation
x)21v(x)1-2 dx
[0, )] -+ [), Ix] by
dx
(Ix
x)2lv(x)1-2 dx.
(18)
M. NASYROVA and V. STEPANOV
234
Put
f+fl
f0
and define fz(x)
O, x
t [k, tt] and
/2(x)
-1
/92
(19)
(x)Iv(x) 12
Then
Now, put
fo+ f2.
f
Then (17) implies
Let us show that
) fc (fy )
dy
dy.
Indeed, applying (18) and (19) we find
cx
Y
f
dy
(lZ
Y) f2(Y) dy
(lZ
sfo(s)ds=foZ(fyZf)
pz(s)) f2(P2(s)) dp2(s)
dy.
37
The lemma implies that satisfies (14) and, consequently, admissible for
(13). Since functions f0 form a cone used for the lower bound in the proof
of Theorem 2.1, we obtain
C
sup (Aa;(o,rx),u,v + Dr + Al.(rx,z.),u,(x_vx)-lv(x) -t-- Bl.(rx,z.),(x-rx)u(x),v)
>> )>o
sup (A1;(rx,)),u,(x-rx)-v(x) + Bl’(rx,.),(x-rDu(x),v).
>> )>0
WEIGHTED HARDY INEQUALITIES
235
For the remaining part of lower bound we suppose, that 0 <
and choose/z (), a) such that
(0"
x)21v(x)1-2 dx
x)21v(x)1-2 dx.
(a
- -
co. Let f L2,v, supp
co, whereas a
Obviously,/z
f(x) >_ O. Define the function p [;,/z] -+ [/z, a] such that
(a
x )2 Iv(x)l -2 dx
)v < a < co
(a
f
[;k,/z],
x)2lv(x)1-2 dx
(s)
and define
fl (x)
0, x
[/x, a] and
=
fl (X)
(/Z, o’).
X
(a -p-l(x))lv(x)l 2
Put
Then it is routine to verify that
Ilfvll 2llfvll22
and
fo(foYf)
dy-O.
Applying Theorem 1.1 we find
-
C
>> A2;(;,z),u,v,
0<
.
< lZ < co.
co and then ;k --+ 0, we obtain the remaining part of the lower
Letting a
bound.
Theorem 2.4 is proved.
CASE (2.5). We have
Fu 112 _<
c F"v
2,
F’(0)
F(co)
0.
(20)
M. NASYROVA and V. STEPANOV
236
THEOREM 2.5
by (10).
Proof
The estimate for the least constant C in (20) is established
Since
{F" F’(0)
0}
F(c)
{F" F’(0)
F(cx)
F’(cx)
0},
we find by Theorem 2.2, that
C
>> B2;(r,oo),u,v + Dl,r + A1;(o,r),(r-x)u(x),v + B1;(o,r),u,(r-x)-lv(x).
Now, suppose
B2;(r,oo),u,v
+ Dl,r + Al’(O,r),(r-x)u(x),v + Bl’(O,r),u,(r-x)-v(x)
<
(x)
and, consequently,
(x-s Iv(x)l
dx < oo, s > r.
Hence, the function
l(s)
(x
s)F"(x) dx
is defined for all s > 0,/ (cxa) 0 and P"
by Theorem 2.2 we obtain the upper bound.
F". It implies that P
F and
CASE (2.1). Now we have
IIFull2
C
[[F"v[[ 2
F(0)= F(oo)- 0.
(21)
Analogously to the case (2.5) we prove the following.
THEOREM 2.6 The estimate for the least constant C in (21) is established
by (12).
CASE (4.1). Now the problem is to characterize
Ilgull2
c liE"vii2,
F(0)= F’(0)= F(cx)- F’(c)= 0.
(22)
WEIGHTED HARDY INEQUALITIES
237
The estimate for the least constant C in (22) is established
The upper bound immediately follows from Theorem 2.4. For
the lower bound suppose (22) is valid. Then it is valid for the weight
re(x) (1 + ex)lv(x)[ instead of v(x). It is easy to see that
Proof
{F"
IlF’Zvell2 < cx3,
{F"
F(0)
IIF’vll2 <
F’(0)
F(cxz)- F’(c)- 0}
, F(0)= F’(0)= F(cx)- 0}.
Consequently, by Theorem 2.4 we have
C
>> A2;(a,),u,v + sup
(Al.(zx,)O,u,(x-rz)-,v(x) + Bl’(rx,)),(x-rz)u(x),v).
)>0
The Fatou theorem brings the required lower bound.
Remark 2.2 The characterization of cases (2.1), (2.5) and (4.1) on a finite
interval is different (for (2.1) and (2.5) see [6]) and (4.1) is so far unknown.
Acknowledgements
The authors wish to thank Professors Hans Triebel and Gordon Sinnamon
for valuable discussion of the paper.
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M. NASYROVA and V. STEPANOV
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